| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 8-s + 6·9-s − 2·11-s − 3·12-s − 5·13-s + 16-s + 6·17-s − 6·18-s + 2·22-s + 2·23-s + 3·24-s + 5·26-s − 9·27-s + 6·29-s + 10·31-s − 32-s + 6·33-s − 6·34-s + 6·36-s + 10·37-s + 15·39-s − 2·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.603·11-s − 0.866·12-s − 1.38·13-s + 1/4·16-s + 1.45·17-s − 1.41·18-s + 0.426·22-s + 0.417·23-s + 0.612·24-s + 0.980·26-s − 1.73·27-s + 1.11·29-s + 1.79·31-s − 0.176·32-s + 1.04·33-s − 1.02·34-s + 36-s + 1.64·37-s + 2.40·39-s − 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14551064309102, −12.60091057486514, −12.14758882864014, −11.79822953240933, −11.67628824573823, −10.83200558074174, −10.49452148152372, −10.04523872221619, −9.853731480527089, −9.305450903573869, −8.485565036675675, −7.848246701922250, −7.691556895673293, −6.977414378862058, −6.603845098399289, −6.100409782583568, −5.559356568058209, −5.099056665597512, −4.696935486668941, −4.179604146028327, −3.133576932295985, −2.731080762560503, −1.981778516184472, −1.017492679459282, −0.8139929797770584, 0,
0.8139929797770584, 1.017492679459282, 1.981778516184472, 2.731080762560503, 3.133576932295985, 4.179604146028327, 4.696935486668941, 5.099056665597512, 5.559356568058209, 6.100409782583568, 6.603845098399289, 6.977414378862058, 7.691556895673293, 7.848246701922250, 8.485565036675675, 9.305450903573869, 9.853731480527089, 10.04523872221619, 10.49452148152372, 10.83200558074174, 11.67628824573823, 11.79822953240933, 12.14758882864014, 12.60091057486514, 13.14551064309102
